Quantum computer and quantum computing method

ABSTRACT

According to an embodiment, a quantum computer includes physical systems X i , a physical system Y j  and a light source unit. The physical systems X i  and the physical system Y j  are provided in a cavity. Each physical system X i  includes states |0&gt; i , |1&gt; i , |2&gt; i  and |e&gt; i , the states |0&gt; i  and |1&gt; i  being used for a qubit, a |2&gt; i -|e&gt; i  transition being resonant with a cavity mode of the cavity. The physical system Y j  includes states |2&gt;′ j  and |e&gt;′ j , a |2&gt;′ j -|e&gt;′ j  transition being resonant with the cavity mode. The light source unit applies laser beams to the cavity to manipulate states of two of physical systems X i , the laser beams including a laser beam for collecting population in the state |2&gt;′ j  in the |2&gt;′ j -|e&gt;′ j  transition.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a Continuation Application of PCT Application No. PCT/JP2014/081415, filed Nov. 27, 2014 and based upon and claiming the benefit of priority from Japanese Patent Application No. 2014-006049, filed Jan. 16, 2014, the entire contents of all of which are incorporated herein by reference.

FIELD

Embodiments described herein relate to a quantum computer utilizing the coupling between a cavity and a physical system.

BACKGROUND

In recent years, researches are being made of a quantum computer which performs a computation by using quantum-mechanical superposition. As one of quantum computers, a quantum computer based on frequency domain quantum computation that discriminates between qubits (quantum bits) in frequency domains is known in the art. In the frequency domain quantum computation, qubits are not discriminated in terms of their positions. Therefore, even qubits that are not to be manipulated are under the effect of operation light with detuning, causing undesired interaction. The undesired interaction may cause gate errors. The adverse effects caused by the undesired interaction may decrease if the frequency difference of the transitions used for qubits is very large. Where the transition distributed in a finite frequency domain is used, however, transitions of small frequency differences may have to be used to load a large number of qubits. Even where the transitions of small frequency differences are used, a quantum computer required is a computer based on the frequency domain quantum computation capable of performing a quantum gate while suppressing the effects caused by the undesired interaction.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 illustrates frequency domain quantum computation.

FIG. 2 illustrates undesired interaction in the physical systems used for the frequency domain quantum computation.

FIG. 3 illustrates a quantum computer according to an embodiment.

FIG. 4 illustrates a physical system which is used for the frequency domain quantum computation in the embodiment.

FIG. 5 illustrates part of the energy states of the Pr³⁺ ion in the Y₂SiO₅ used in the embodiment.

FIG. 6 is a flowchart illustrating a method for manipulating two physical systems according to the embodiment.

FIG. 7 is a waveform chart illustrating pulses used for the adiabatic passage via a cavity according to the embodiment.

FIG. 8 illustrates a quantum computer according to another embodiment.

FIG. 9 illustrates a quantum computer according to still another embodiment.

DETAILED DESCRIPTION

According to one embodiment, a quantum computer includes physical systems X_(i) (i=1, 2, . . . , N; N being an integer not less than 2), a physical system Y_(j) (j=1, 2, . . . , N₂; N₂ being an integer not less than 1), and a light source unit. The physical systems X, are provided in a cavity, each of the physical systems X_(i) comprising at least four states including states |0>_(i), |1>_(i), |2>_(i) and |e>_(i), energy of the state |e>_(i) being higher than energy of the states |0>_(i) and |1>_(i) used for a qubit and energy of the state |2>_(i) used for assisting a gate operation, a |2>-|e>_(i) transition being resonant with a cavity mode of the cavity. The physical system Y_(j) is provided in the cavity, the physical system Y_(j) being different from the physical systems X_(i), the physical system Y_(j) comprising at least two energy states including states |2>′_(j) and |e>′_(j), energy of the state |e>′_(j) being higher than energy of the state |2>′_(j), a |2>′_(j)-|e>′_(j) transition being resonant with the cavity mode. The light source unit applies laser beams to the cavity to manipulate a state of a physical system X_(a) (where s is a natural number not more than N) and a state of physical system X_(t) (where t is a natural number not more than N and different from s) of the physical systems X_(i), the laser beams including a first laser beam resonating with a |1>_(s)-|e>_(s) transition, a second laser beam resonating with a |1>_(t)-|e>_(t) transition, and a third laser beam for collecting, population in the state |2>′_(j) in the |2>′_(i)-|e>′_(j) transition of the physical system Y.

Hereinafter, various embodiments will be described with reference to the drawings. In the following embodiments, the like elements will be denoted by the like reference symbols, and redundant descriptions will be omitted where appropriate.

First of all, a description will be given of frequency domain quantum computation and “undesired interaction” which may occur at the quantum gate thereof. A description will also be given of the “resonance condition” under which the effects of the undesired interaction are particularly marked. Further, a description will be given of a quantum gate operation method and configuration, which control the resonance condition so as to eliminate the effects of the undesired interaction.

[Frequency Domain Quantum Computation]

In the frequency domain quantum computation, a plurality of physical systems, which are arranged in an optical cavity (also called an optical resonator) and each of which has a transition resonating with a common cavity mode (eigenmode) and has other transitions different in frequency depending upon the physical systems, are used as qubits. For example, ions and atoms can be used as the physical systems. In the frequency domain quantum computation, each physical system can be selectively manipulated by radiating a laser beam resonating with the transition frequency of the physical system.

A description will be given of the case where N four-level systems X_(i) (i=1, 2, 3, . . . , N) are used as physical systems. It is noted here that N is an integer which is not less than 2. Each of the four-level systems X_(i) has four energy states. The four energy state will be expressed as |0>_(i), |1>_(i), |2>_(i), and |e>_(i) in the energy level ascending order. The suffix i attached to each state (each ket vector) is for identifying the four-level systems X_(i). In the following, the suffix i may be omitted. The states |9>_(i) and |1>_(i) are used for a qubit, and the state |2>_(i) is used for assisting the gate operation. The excited state |e>_(i) has a higher level of energy than the states |0>_(i), |1>_(i) and |2>_(i). The |2>_(i)-|e>_(i) transition (namely, a transition between the state I2>_(i) and the state |e>_(i)) is a transition resonating with a common cavity mode of the cavity. The frequency of the |1>_(i)-|e>_(i) transition differs depending upon the four-level systems X_(i).

FIG. 1 illustrates a physical system that is used for the frequency domain quantum computation when N=3. As shown in FIG. 1, physical systems X₁, X₂ and X₃ are coupled to a common cavity mode. To be specific, the |2>_(i)-|e>₁ transition, the |2>₂-|e>₂ transition and the |2>₃-|e>₃ transition resonate with the common cavity mode. That is, the frequency of the |2>₁-|e>₁ transition, the frequency of the |2>₂-|e>₂ transition and the frequency of the |2>₃-|e>₃ transition are equal to the resonance frequency of the common cavity mode. The frequency of the |1>_(i)-|e>₁ transition, the frequency of the |1>₂-|e>₂ transition and the frequency of the |1>₃-|e>₃ transition differ from each other.

Where two of these four-level systems X_(i) are manipulated (for example, the states of four-level systems X₁ and X₂ are manipulated), operation light L₁ resonating with the |1>₁-|e>₁ transition and operation light L₂ resonating with the |1>₂-|e>₂ transition are radiated to all four-level systems X_(i). Ideally, the states of four-level systems X₁ and X₂ can be selectively manipulated by the radiation of operation light L1 and operation light L2. As a method for manipulating the states used in a quantum gate, an adiabatic passage via a cavity is known in the art. In this method, when, for example, the state of the four-level system X₁ is manipulated from the initial state |1>₁ to the state |2>₁ and the state of the four-level system X₂ is manipulated from the initial state |2>₂ to the state |1>₂, the intensities of the operation light L₁ and L₂ are controlled in such a manner that the Rabi frequencies Ω₁ and Ω2 of the operation light L₁ and L₂ satisfy Expression (1) below under the condition where τ₁>τ₂.

$\begin{matrix} {{{\Omega_{1}(t)} = {\Omega_{0}{\exp \left\lbrack {- \frac{\left( {t - \tau_{1}} \right)^{2}}{2\sigma^{2}}} \right\rbrack}}}{{\Omega_{2}(t)} = {\Omega_{0}{\exp \left\lbrack {- \frac{\left( {t - \tau_{2}} \right)^{2}}{2\sigma^{2}}} \right\rbrack}}}} & (1) \end{matrix}$

[Undesired Interactions]

Strictly speaking, undesired interactions described below inevitably occur in the frequency domain quantum computation. Since the four-level systems X_(i) are not discriminated from one another in light of their positions, operation light with detuning is applied to four-level systems X_(i) which are not to be manipulated, leading to the undesired interactions. To be specific, the undesired interactions include interactions caused by operation light L₁ in four-level systems X₂ and X_(j) (j=3, 4, . . . , N) and interactions caused by operation light L₂ in four-level systems X₁ and X_(j) (j=3, 4, . . . , N). FIG. 2 illustrates a physical system including undesired interactions when N=3. In FIG. 2, the solid-line two-headed arrow connecting the state |1>_(i) and the state |e>_(i) indicates a necessary interaction, and a broken-line two-headed arrow indicates an undesired interaction. The thick solid-line two-headed arrow connecting the state |2>_(i) and the state |e>_(i) indicates a cavity mode.

If the frequency difference between the |1>_(i)-|e>_(i) transitions of the four-level systems X_(i) is very large, the undesired interaction does not cause a significant effect. However, since the frequencies of the transitions are distributed in a finite frequency domain, transitions of a small frequency difference should be utilized to use a large number of qubits. For this reason, it is desirable that the quantum gate can be performed with high efficiency even if the frequency differences are small.

[Resonance Condition]

A description will be given of the property of an undesired interaction. The Hamiltonian describing a physical system used for the frequency domain quantum computation including an undesired interaction (such as the physical system depicted in FIG. 2) is expressed by Expression (2) set forth below.

$\begin{matrix} {{{H(t)}/\hslash} = {{\sum\limits_{i = 1}^{N}\; \left\lbrack {{\sum\limits_{{j = 0},1,2,e}\; {\omega_{j}^{(i)}\sigma_{jj}^{(i)}}} - {\gamma\sigma}_{ee}^{(i)}} \right\rbrack} + \left\lbrack {{2\pi \; f_{g}a^{\dagger}a} - {\; \kappa \; a^{\dagger}a}} \right\rbrack + {\sum\limits_{i = 1}^{N}\; \left\lbrack {{{ga}\; \sigma_{e\; 2}^{(i)}} + {{\Omega_{1}(t)}^{{- }\; 2\; \pi \; f_{1}t}\sigma_{e\; 1}^{(i)}} + {{\Omega_{2}(t)}^{{- }\; 2\; \pi \; f_{2}t}\sigma_{e\; 1}^{(i)}} + {H.c.}} \right\rbrack}}} & (2) \end{matrix}$

where σ^((i)) _(ab) is an operator for transition of state |b>_(i) of four-level system X_(i) to state |a>_(i), a and

are the annihilation operator and the creation operator of the cavity mode, respectively, g is a coupling constant for coupling between the cavity mode and a physical system, γ is a relaxation rate of transition, κ is an attenuation constant of the cavity, and H.c. is a Hermitian conjugate.

The terms of the Hamiltonian will be described. The first term includes an energy term of each state of each ion and an energy relaxation term of each ion. The second term includes an energy term and a cavity relaxation term of the cavity mode. In the following, a description will be given of the case where γ=κ=0 on the assumption of an ideal strongly-coupled system. The third term is an interaction term, including an interaction between the cavity mode and the |2>_(i)-|e>_(i) transition of each four-level system X_(i), an interaction between the operation light L₁ and the |1>_(i)-|e>_(i) transition of each four-level system X_(i), and an interaction between the operation light L₂ and the |1>_(i)-|e>_(i) transition of each four-level system X_(i).

The interaction Hamiltonian in the Hamiltonian given by Expression (2) is expressed as Expression (4) below, in which H₀ given by Expression (3) below is used.

$\begin{matrix} {{{H_{0}/\hslash} \equiv {{\sum\limits_{i = 1}^{N}\; \left\lbrack {\sum\limits_{{j = 0},1,2,e}\; \left( {\omega_{j}^{(i)}\sigma_{jj}^{(i)}} \right)} \right\rbrack} + {2\pi \; f_{g}a^{\dagger}a}}}\begin{matrix} {{{H^{\prime}(t)}/\hslash} = {{\exp \left\lbrack {{- }\; H_{0}{t/\hslash}} \right\rbrack}\left( {H/\hslash} \right){\exp \left\lbrack {\; H_{0}{t/\hslash}} \right\rbrack}}} \\ {= {{\sum\limits_{i = 1}^{N}\; {{ga}\; \sigma_{e\; 2}^{(i)}}} + {{\Omega_{1}(t)}\left\{ {\sigma_{e\; 1}^{(1)} + {^{{- {\Delta}}\; t}\sigma_{e\; 1}^{(2)}} + {\sum\limits_{j = 3}^{N}\; {^{{- {\Delta}_{j}}t}\sigma_{e\; 1}^{(j)}}}} \right\}} +}} \\ {{{{\Omega_{2}(t)}\left\{ {{^{\; \Delta \; t}\sigma_{e\; 1}^{(1)}} + \sigma_{e\; 1}^{(2)} + {\sum\limits_{j = 3}^{N}\; {^{{- {{({\Delta + \Delta_{j}})}}}t}\sigma_{e\; 1}^{(j)}}}} \right\}} + {H.c.}}} \end{matrix}} & (3) \end{matrix}$

In Expression (4), the following relations are used.

(ω_(e) ⁽¹⁾−ω₁ ⁽¹⁾)−2πƒ₁=0, (ω_(e) ⁽¹⁾−ω₁ ⁽¹⁾)−2πƒ₂=Δ, (ω_(e) ⁽²⁾−ω₁ ⁽²⁾)−2πƒ₁=−Δ,

(ω_(e) ⁽²⁾−ω₁ ⁽²⁾)−2πƒ₂=0, (ω_(e) ^((j))−ω₁ ^((j)))−2πƒ₁=Δ_(j), (ω_(e) ^((j))−ω₁ ^((j)))−2πƒ₂=Δ+Δ_(j),

(ω_(e) ^((i))−ω₂ ^((i))−)2πƒ_(g)=0, for i=1,2, . . . N and j=3,4, . . . , N.

Δ is a frequency difference between the |e>₁−|1>_(j) transition and the |e>₂-|1>₂ transition, and Δ_(j) is a frequency difference between the |e>₁-|1>₁ transition and the |e>_(j)-|1>_(j) transition.

The interaction Hamiltonian given by Expression (4) is divided into a necessary interaction H₁ and an undesired interaction V.

$\begin{matrix} {{H^{\prime}(t)} = {{H_{1}(t)} + {{V(t)}\left\{ \begin{matrix} {{{H_{1}(t)}/\hslash} = {{\sum\limits_{i = 1}^{N}\; {{ga}\; \sigma_{e\; 2}^{(i)}}} + {{\Omega_{1}(t)}\sigma_{e\; 1}^{(1)}} + {{\Omega_{2}(t)}\sigma_{e\; 1}^{(2)}} + {H.c.}}} \\ {{{V(t)}/\hslash} = {{{\Omega_{1}(t)}\left\{ {{^{{- }\; \Delta \; t}\sigma_{e\; 1}^{(2)}} + {\sum\limits_{j = 3}^{N}\; {^{{- }\; \Delta_{j}t}\sigma_{e\; 1}^{(j)}}}} \right\}} +}} \\ {{{\Omega_{2}(t)}\left\{ {{^{\; \Delta \; t}\sigma_{e\; 1}^{(1)}} + {\sum\limits_{j = 3}^{N}\; {^{{- {{({\Delta + \Delta_{j}})}}}t}\sigma_{e\; 1}^{(j)}}}} \right\}} + {H.c.}} \end{matrix} \right.}}} & (5) \end{matrix}$

In the adiabatic passage via the cavity, the error probability caused by the undesired interactions can be calculated by perturbation theory, using Expression (5). An initial state |ψ(0)

is assumed to be a dark state |ψ(0)

that is one of eigenstates of H₁. The error probability in the adiabatic passage is the probability of transition to another eigenstate |ψ_(n)(t)

(n≠0) at time t. In order to perform this manipulation with high efficiency, it is general to use operation light L₁ and operation light L₂ that satisfy Ω_(1,2)<<g. Since V is smaller than H₁ in this case, the time evolution can be calculated by performing perturbation theory with respect to V, as shown in Expression (6) set forth below.

$\begin{matrix} {{{\left. {{\left. {{{{\left| {\psi (t)} \right.\rangle} = \left| {\psi_{0}(t)} \right.}\rangle} + {\sum\limits_{n}\; C_{n}^{(1)}}} \middle| {\psi_{n}(t)} \right.\rangle} + {\sum\limits_{n}\; C_{n}^{(2)}}} \middle| {\psi_{n}(t)} \right.\rangle} + {O\left( V^{3} \right)}}\left\{ \begin{matrix} {{C_{n}^{(1)}(t)} = {\frac{1}{\hslash}{\int_{0}^{t}\ {{t^{\prime}}^{{- {({E_{n} - E_{0}})}}{t^{\prime}/{\hslash}}}{\langle{{\psi_{n}\left( t^{\prime} \right)}{{V\left( t^{\prime} \right)}}{\psi_{0}\left( t^{\prime} \right)}}\rangle}}}}} \\ {{C_{n}^{(2)}(t)} = {\left( \frac{1}{\hslash} \right)^{2}{\int_{0}^{t}\ {{t^{\prime}}{\int_{0}^{t^{\prime}}\ {{t^{''}}^{{- {({E_{n} - E_{k}})}}{t^{\prime}/{\hslash}}}^{{- {({E_{k} - E_{0}})}}{t^{\prime}/{\hslash}}}}}}}}} \\ {{\langle\left. {\psi_{n}\left( t^{\prime} \right)} \middle| {V\left( t^{\prime} \right)} \middle| {\psi_{k}\left( t^{\prime} \right)} \right.\rangle}{\langle{{\psi_{k}\left( t^{''} \right)}{{V\left( t^{''} \right)}}{\psi_{0}\left( t^{''} \right)}}\rangle}} \end{matrix} \right.} & (6) \end{matrix}$

In Expression (6), E_(n) is an eigenvalue corresponding to the eigenstate |ψ_(n)

of H₁. E_(n) changes with time in accordance with Ω_(1,2) alone. In the case where Ω_(1,2)<<g, changes in E_(n) are sufficiently small, compared with the absolute value of E_(n). Therefore, when the exponents of the exponent functions in coefficients C_(n) ⁽¹⁾ and C_(n) ⁽²⁾ are zero, the error probability increases to such an extent that the perturbation theory becomes invalid. In view of the fact that V(t′) contains an oscillation term, the resonance condition given by Expression (7) below can be obtained from the first-order coefficient C_(n) ⁽¹⁾ as a condition under which the error probability of the adiabatic passage via the cavity increases.

(E _(n) −E ₀)/

=±Δ,±Δ_(j),±(Δ+Δ_(j))  (7)

Likewise, the resonance condition given by Expression (8) can be obtained from the second-order coefficient.

$\begin{matrix} \left\{ \begin{matrix} {{{\left( {E_{k} - E_{0}} \right)/\hslash} = {\pm \Delta}},{\pm \Delta_{j}},{\pm \left( {\Delta + \Delta_{j}} \right)}} \\ {{{\left( {E_{n} - E_{k}} \right)/\hslash} = {\pm \Delta}},{\pm \Delta_{j}},{\pm \left( {\Delta + \Delta_{j}} \right)}} \end{matrix} \right. & (8) \end{matrix}$

Compared with the resonance condition obtained from the first-order coefficient C_(n) ⁽¹⁾, the resonance condition obtained from the second-order coefficient C_(n) ⁽²⁾ does not contribute much. However, when the two or more conditions are satisfied simultaneously, the contribution is significantly large. As a condition for permitting two or more conditions to be satisfied in the resonance condition of Expression (8), the following condition given by Expression (9) is obtained.

(E _(n) −E ₀)/

=±Δ,±Δ_(j),±2Δ,±2Δ_(j),±(Δ−Δ_(j)),±(2Δ+Δ_(j)),±(Δ+2Δ_(j)),±2(Δ+Δ_(j))   (9)

for j=3, 4, . . . , N.

The analytical solution to the resonance condition can be obtained by calculating the eigenvalue E_(n) of H₁. Where the term of Ω_(1,2) of H₁ is set at zero for the case where Ω_(1,2)<<g, the eigenvalue E_(n) can be obtained based on the analogy of well-known vacuum Rabi splitting. The number of four-level systems X_(i) having a population in state |2>_(i) or state |e>_(i) will be denoted by N₂, the number of four-level systems X_(i) having a population in state |e>_(i) will be denoted by n_(e), and the number of photons in the cavity mode will be denoted by n_(c). The eigenvalues of H₁ can be classified by designating the total number N_(e) of excitons (N_(e)=n_(e)+n_(c)) and a maximal value n_(e)|_(max) of n_(e). Part of the eigenvalues of H₁ are given by Expression (10) below.

E _(1,N) ₂ /

=±√{square root over (N ₂)}g(N _(e)=1,n _(e)|_(max) =N ₂)

E _(2,N) ₂ /

=0,±√{square root over (4N ₂=2)}g(N _(e)=2,n _(e)|_(max) =N ₂)

E _(1,N) ₂ ⁻¹/

=±√{square root over (N ₂−2)}g(N _(e)=2,n _(e)|_(max) =N ₂−1)  (10)

By using Expressions (7), (9) and (10), the resonance condition under which undesired interactions increase can be analytically obtained. The resonance condition should be avoided in order to perform the adiabatic passage via the cavity with high efficiency, namely, to perform the quantum gate efficiently in the frequency domain quantum computation.

[Control of Resonance Condition]

As shown in Expression (10), the eigenvalues of H₁ vary in accordance with the number N₂ of four-level systems X_(i) having a population in either state |2>_(i) or state |e>_(i).

This means that the resonance condition can be controlled based on the number of four-level systems X_(i) having a probability amplitude in the transition coupled to the cavity. This kind of control is referred to as a control based on the addition of the transition coupled to the cavity.

For example, in a quantum computer utilizing three qubits, the adiabatic passage via the cavity is performed such that four-level systems X₁, X₂ and X₃ are manipulated from the initial states |1>₁, |2>₂ and |1>₃ to states |2>₁, |1>₂ and |1>₃. In the manipulation, where N_(e) is equal to 1, N₂ is equal to 2, and where N_(e) is equal to 2, N₂ is equal to 3. Therefore, the eigenvalues of H₁ can be obtained based on Expression (10), in the way expressed in Expression (11). The resonance condition can be obtained from Expressions (7) and (9), using the eigenvalues represented by Expression (11).

E _(1,2)=±√{square root over (2)}g

E _(2,3)=0,±√{square root over (10)}g

E _(2,2) =±g  (11)

Where three physical systems having a transition probability in the transition coupled to the cavity are added, the eigenvalues of H₁ can be obtained in the way indicated by Expression (12) below.

E _(1,2)=±√{square root over (5)}g

E _(2,3)=0,±√{square root over (22)}g

E _(2,2)=±2g  (12)

In this case, the region in which the resonance condition can be avoided significantly increases in the region where |Δ|<g and |A_(j)|<g. Therefore, if a frequency distribution of the four-level systems X_(i) is stochastically provided, the probability that the quantum gate can be performed with high efficiency increases. In addition, more qubits can be used.

As described above, the undesired interactions can be suppressed by controlling the resonance condition by adding a transition coupled to the cavity, and as a result, frequency domain quantum computation can be performed with high efficiency. Please note that what is described herein is nothing but an example. With respect to the frequency distribution of physical systems actually provided, the number of additional transitions coupled to the cavity can be changed in such a manner that control to a more appropriate condition is enabled.

Hereinafter, embodiments will be described with reference to FIGS. 3 to 9.

First Embodiment

FIG. 3 schematically illustrates a quantum computer 300 according to the first embodiment. According to the first embodiment, Pr³⁺ ions doped in Y₂SiO₅ crystal are used as specific physical systems that can be regarded as four-level system X_(i). As shown in FIG. 3, the quantum computer 300 includes a sample 315, which serves as a cavity wherein dielectric multi-layer mirrors are arranged on sides of the Pr³⁺-ion doped Y₂SiO₅ crystal (Pr:YSO). The sample 315 is arranged inside a cryostat 316 and kept at a low temperature (e.g., at 4K).

Let us consider the case where, as shown in FIG. 4, six Pr³⁺ ions are used in the frequency domain quantum computation and three of these Pr³⁺ ions are used as qubits. The three Pr³⁺ ions used as qubits will be expressed as X₁, X₂ and X₃, and the remaining three Rr³⁺ ions will be expressed as Y₁, Y₂ and Y₃. The ions Y₁, Y₂ and Y₃ are used for controlling the resonance condition.

To be specific, each of the ions X₁, X₂ and X₃ has states |0>, |1>, |2>and |e> in the ascending order of energy level. The states |0>_(i) and |1>_(i) are used for the qubit, and the state |2>_(i) is used for assisting the gate operation. The |2>-|e> transitions of the ions X₁, X₂ and X₃ resonate with the common cavity mode of the cavity. Each of the ions Y₁, Y₂ and Y₃ has states |0>′, |1>′, 12>′ and |e>′ in the ascending order of energy level. In FIG. 4, illustration of the states |0>′ and |1>′of each of the ions Y₁, Y₂ and Y₃ is omitted. The |2>′-|e>′ transitions of the ions Y₁, Y₂ and Y₃ resonate with the common cavity mode of the cavity. The prime “′” is attached to the states of the ions Y₁, Y₂ and Y₃. The addition of the prime is intended to mean that the physical systems used for controlling the resonance conditions and the physical systems used as qubits may be different in kind. The ions Y₁, Y₂ and Y₃ may be the same as the ions X₁, X₂ and X₃ (Pr³⁺ ions in this example). Alternatively, they may be different from the ions X₁, X₂ and X₃ in kind, or may be a combination of different kinds of ions. Where the ions Y₁, Y₂ and Y₃ are different in kind from the ions X₁, X₂ and X₃, the ions Y₁, Y₂ and Y₃ do not have to have state |0>′ or |1>′.

FIG. 5 illustrates part of the energy states of the Pr³⁺ ion in the Y₂SiO₅ crystal. In FIG. 5, three (|±1/2>, |=3/2> and |±5/2>) of the hyperfine structure states of a ground state ³H₄ are shown, and three (|±1/2>, |=3/2> and |±5/2>) of the hyperfine structure states of an excited state ¹D₂ are shown. The wavelength corresponding to the transition frequency between the excited state ¹D₂ and the ground state ³H₄ is approximately 606 nm. The hyperfine structure states |±5/2>, |±3/2> and |7E1/2> of the ground state ³H₄ correspond to the states |0>, |1> and |2> shown in FIG. 4, and the hyperfine structure state |5/2> of the excited state ¹D₂ corresponds to the state |e> shown in FIG. 4. In this case, the |0>-|e> transition, the |1>-|e> transition and the |2>-|e> transition are optically allowed, and the |0>-|1> transition, the |0>-|2> transition and the |1>-|2> transition are optically forbidden.

In the quantum computer 300 shown in FIG. 3, the ring dye laser 302 excited by the argon ion laser 301 is used as a light source. A laser beam emitted from the ring dye laser 302 is split into three laser beams by means of two beam splitters 303 and 304, and these three laser beams are guided to acousto-optic modulators 306, 307 and 308, respectively. To be specific, the beam splitter 303 splits the laser beam emitted from the ring dye laser 302 into two laser beams, and one of them enters the acousto-optic modulator 306 and the other is directed to the beam splitter 304. The beam splitter 304 splits the incident laser beam into two laser beams, and one of them enters the acousto-optic modulator 307 and the other is reflected by a mirror and then enters the acousto-optic modulator 308.

The acousto-optic modulators 306, 307 and 308 modulate the incident laser beams in accordance with a signal generated by a controller 309, thereby generating modulated laser beams 351, 352 and 353, respectively. The modulated laser beam 351 is guided to the sample 315 by mirrors 310 and 311 and lens 314. The modulated laser beam 352 is guided to the sample 315 by the lens 314. The modulated laser beam 353 is guided to the sample 315 by mirrors 312 and 313 and the lens 314. In the present embodiment, the argon ion laser 301, the ring dye laser 302, the beam splitters 303 and 304, the mirror 305, the acousto-optic modulators 306 to 308, the mirrors 310 to 313 and the lens 314 jointly form the light source unit 320.

A method for manipulating the ions X₁ and X₂, which are qubits included in the three qubits, will be specifically described, referring to FIGS. 3, 6 and 7. First, the light source unit 320 applies the modulated laser beam 353 to the sample 315, so that a population is collected in state |2> of transition |2>-|e>, with respect to the ions Y₁, Y₂ and Y₃ (step S601 in FIG. 6). To be specific, the acousto-optic modulator 308 modulates the incident laser beam such that the modulated laser beam 353 resonates with the |0>-|e> transition or the |1>-|e> transition.

Subsequently, in the state where the modulated laser beam 353 is kept applied, the light source unit 320 simultaneously applies the modulated laser beams 351 and 352, which are for manipulating the ions X₁ and X₂, to the sample 315 (step S602 in FIG. 6). The “simultaneous application” is intended to mean that the application time of the modulated laser beam 351 and the application time of the modulated laser beam 352 overlap each other at least in part. To be specific, the acousto-optic modulator 306 modulates the incident laser beam such that the Rabi frequency Ω₁ of the modulated laser beam 351 changes in the way shown in Expression (1), and the acousto-optic modulator 307 modulates the incident laser beam such that the Rabi frequency Ω₂ of the modulated laser beam 352 changes in the way shown in Expression (1). When the states of the ions X₁ and X₂ are changed from the initial states |1>₁ and |2>₂ to the states |2>₁ and |1>₂, τ₁ and τ₂ are made to satisfy τ₁>τ₂. FIG. 7 illustrates temporal changes in the Rabi frequencies Ω₁ and Ω₂ when the ions X₁ and X₂ is manipulated from the initial states |1>₁ and |2>₂ to the states |2>₁ and |1>₂. In FIG. 7, the transverse axis represents time, while the vertical axis represents Rabi frequencies. The waveform indicated by the dashed-dotted line is Rabi frequency Ω₁, and the waveform indicated by the solid line is Rabi frequency Ω₂. As an example, the parameters of the Rabi frequencies Ω₁ and Ω₂ of the laser beams 351 and 352 are set as follows: Ω₀=1 kHz, τ₁=64.1 ms, τ₂=55.9 ms and σ→20 ms.

The modulated laser beams 351, 352 and 353 are applied to the sample 315 in this manner, and this enables the ions the X₁ and X₂ to be changed from the initial states |1>₁ and |2>2 to the state |2>₁ and |1>₂, and yet the state of the ions X₃ (e.g., |1>₃) is kept unchanged while avoiding the resonance condition.

As described above, the quantum computer according to the first embodiment uses a physical system which is different from physical systems used as qubits and which include a transition coupling with the common cavity mode. By using such physical systems, the quantum gate can be performed very efficiently, while suppressing the undesired interactions.

Second Embodiment

In the first embodiment, one light source is used for generating laser beams used for state manipulations. In the second embodiment, a light source is prepared for each of laser beams used for state manipulations.

FIG. 8 schematically illustrates a quantum computer 800 according to the second embodiment. As shown in FIG. 8, the quantum computer 800 includes semiconductor lasers 801, 802 and 803. Laser beams emitted from the semiconductor lasers 801, 802 and 803 are guided to acousto-optic modulators 306, 307 and 308, respectively. Since the operation of the acousto-optic modulators 306, 307 and 308 is similar to that described in the first embodiment, the explanation thereof is omitted. In the present embodiment, the semiconductor lasers 801 to 803, the acousto-optic modulators 306-308, the mirrors 310-313 and the lens 314 jointly form a light source unit 820.

Like the first embodiment, the second embodiment enables efficient performance of a quantum gate.

Third Embodiment

FIG. 9 schematically illustrates a quantum computer 900 according to the third embodiment. The quantum computer 900 shown in FIG. 9 is similar in configuration to the quantum computer 300 of the first embodiment shown in FIG. 3. In the following, a description will be given mainly of the features distinguishing the third embodiment from the first embodiment.

In the third embodiment, ions X₁, X₂ and X₃ used as qubits and ions Y₁, Y₂ and Y3 used for the control of the resonance condition are located in different space regions. For example, the ions X₁, X₂ and X₃ are selected from the Pr³⁺ ions in the upper portion of a sample 315, and the ions Y₁, Y₂ and Y₃ are selected from the Pr³⁺ ions in the lower portion of the sample 315. In this case, as shown in FIG. 9, the light source unit 920 applies the modulated laser beams 351 and 352 to the upper portion of the sample 315, and applies the modulated laser beam 353 to the lower portion of the sample 315.

Like the first embodiment, the third embodiment enables efficient performance of a quantum gate.

Fourth Embodiment

In connection with the fourth embodiment, a description will be given of an example of a method for selecting Pr³⁺ ions used as qubits from the Pr³⁺ ions coupling with the common cavity mode. The quantum computer of the fourth embodiment is similar in configuration to the quantum computer 300 (FIG. 3) of the first embodiment and performs the quantum gate in the same method as the first embodiment. Accordingly, a repetitive description will be omitted.

M Pr³⁺ ions included in a sample 315 and including transitions coupling with the common cavity mode will be referred to as ion 1, ion 2, . . . , ion M in the ascending order of the frequency of transition |1>-|e>. It is noted here that M is an integer which is not less than 2. The frequency difference between the |1>₁-|e>₁ transition and the |1>₂-|e>₂ transition will be expressed as Δ, and the frequency difference between the |1>₁-|e>₁ transition and the |1>_(j)-|e>_(j) transition (j: an integer not less than 3 and not more than M) will be expressed as Δ_(j). In this case, such ions as satisfy Ω₀<Δ<N₂g/N and Ω₀<Δ_(j)<N₂g/N are used as qubits. A predetermined number of ions used for the control of the resonance condition are selected in the descending order of the frequency difference

The number N₂ of physical systems (Pr³⁺ ions in this example) used for the control of the resonance condition may be any number not less than 1, but is preferably a value that is at least equal to the square of the number of qubits. This is because the eigenvalues of H₁ are practically dependent on the square root of N₂, as shown in Expression (10). For example, where three qubits are used, nine Pr³⁺ ions including transitions coupling with the common cavity mode are used as physical systems for the control of the resonance condition. The nine Pr³⁺ ions (referred to as ions Y₁ to Y₉) are selected from among the ions 1 to M in the descending order of frequency difference Δ_(i) as in the method described above. In this case, manipulations similar to those performed for the ions Y₁, Y₂ and Y₃ in the first embodiment are performed for the ions Y₁ to Y₉. To be specific, the sample 315 is irradiated with a laser beam that is modulated in such a manner as to enable the population to be collected in state |2> with respect to the ions Y₁ to Y₉.

Like the first embodiment, the fourth embodiment enables efficient performance of a quantum gate.

While certain embodiments have been described, these embodiments have been presented by way of example only, and are not intended to limit the scope of the inventions.

Indeed, the novel embodiments described herein may be embodied in a variety of other forms; furthermore, various omissions, substitutions and changes in the form of the embodiments described herein may be made without departing from the spirit of the inventions. The accompanying claims and their equivalents are intended to cover such forms or modifications as would fall within the scope and spirit of the inventions. 

What is claimed is:
 1. A quantum computer comprising: physical systems X_(i) (i=1, 2, . . . , N; N being an integer not less than 2) provided in a cavity, each of the physical systems X_(i) comprising at least four states including states |0>_(i), |1>_(i), |2>_(i) and |e>_(i), energy of the state |e>_(i) being higher than energy of the states |0>_(i) and |1>i used for a qubit and energy of the state |2>_(i) used for assisting a gate operation, a |2>_(i)-|e>_(i) transition being resonant with a cavity mode of the cavity; a physical system Y_(j) (j=1, 2, . . . , N₂; N₂ being an integer not less than 1) provided in the cavity, the physical system Y_(j) being different from the physical systems X_(i), the physical system Y_(j) comprising at least two energy states including states |2>′_(j) and |e>′_(j), energy of the state |e>′_(j) being higher than energy of the state |2>′_(j), a |2>′_(j)−|e>′_(j) transition being resonant with the cavity mode; and a light source unit which applies laser beams to the cavity to manipulate a state of a physical system X_(s) (where s is a natural number not more than N) and a state of physical system X_(t) (where t is a natural number not more than N and different from s) of the physical systems X_(i), the laser beams including a first laser beam resonating with a |1>_(s)-|e>_(s) transition, a second laser beam resonating with a |1>_(t)-|e>_(t) transition, and a third laser beam for collecting population in the state |2>′_(j) in the |2>′_(j)-|e>′_(j) transition of the physical system Y_(j).
 2. The quantum computer according to claim 1, wherein the light source unit comprises: a light source which emits a laser beam; two beam splitters which divide the laser beam into a first branch laser beam, a second branch laser beam and a third branch laser beam; a first optical modulator which modulates the first branch laser beam to generate the first laser beam; a second optical modulator which modulates the second branch laser beam to generate the second laser beam; and a third optical modulator which modulates the third branch laser beam to generate the third laser beam.
 3. The quantum computer according to claim 1, wherein the light source unit comprises: a first light source; a first optical modulator which modulates a laser beam emitted from the first light source to generate the first laser beam; a second light source; a second optical modulator which modulates a laser beam emitted from the second light source to generate the second laser beam; a third light source; and a third optical modulator which modulates a laser beam emitted from the third light source to generate the third laser beam.
 4. The quantum computer according to claim 1, wherein the cavity comprises a first space region including the physical systems X_(i), and a second space region including the physical system Y_(i), the second space region being different from the first space region, and the light source unit applies the third laser beam to the second space region.
 5. A quantum computing method comprising: preparing physical systems X_(i) (i=1, 2, . . . , N; N being an integer not less than 2) provided in a cavity, each of the physical systems X_(i) comprising at least four states including states |0>_(i), |1>_(i), |2>_(i) and |e>_(i), energy of the state |e>_(i) is higher than energy of the states |0>_(i) and |1>i used for a qubit and energy of the state |2>_(i) used for assisting a gate operation, a |2>_(i)-|e>_(i) transition being resonant with a cavity mode of the cavity; preparing a physical system Y_(j) (j=1, 2, . . . , N₂; N₂ being an integer not less than 1) provided in the cavity, the physical system Y_(j) being different from the physical systems X_(i), physical system Y_(j) comprising at least two energy states including |2>′_(j) and |e>′_(j), energy of the state |e>′_(j) being higher than energy of the state |2>′_(j), a |2>′_(j)-|e>t_(j) transition being resonant with the cavity mode; and applying laser beams to the cavity to manipulate state of a physical system X, (where s is a natural number not more than N) and a state of physical system X_(t) (where t is a natural number not more than N and different from s) of the physical systems X_(i), the laser beams including a first laser beam resonating with a |1>_(s)-|e>_(s) transition, a second laser beam resonating with a |1>_(t)-|e>_(t) transition, and a third laser beam for collecting population in the state |2>′_(j) in the |2>′_(j)-|e> transition of the physical system Y.
 6. The quantum computing method according to claim 5, further comprising: applying the first laser beam and the second laser beam to the cavity, after applying the third laser beam to collect population in the state |2>′_(j) of the physical system Y_(j).
 7. The quantum computing method according to claim 5, wherein if the physical systems X, comprises physical systems X_(i), X₂, . . . , X_(N) in ascending order of frequency of a |1>_(i)-|e>, transition, a frequency difference between a |1>_(i)-|e>₁ transition and a |1>₂-|e>₂ transition is Δ, and a frequency difference between a |1>₁-|e>₁ transition and a |1>_(k)-|e>_(k) transition is L_(k), then the plurality of physical systems X_(i) satisfy: Ω₀<Δ<N₂g/N and Ω₀<Δ_(k)<N₂g/N, where g is a coupling constant between the cavity mode and each of the physical systems X_(i), and Ω₀ is a parameter for determining a maximum value of Rabi frequencies of the first laser beam and the second laser beam.
 8. The quantum computing method according to claim 5, wherein N₂≧N² is satisfied. 